Algorithmic Gauss-Manin Connection

Transcription

1 Algorithmic Gauss-Manin Connection Algorithms to Compute Hodge-theoretic Invariants of Isolated Hypersurface Singularities Mathias Schulze Vom Fachbereich Mathematik der Universität Kaiserslautern zur Verleihung des akademischen Grades Doktor der Naturwissenschaften (Doctor rerum naturalium, Dr. rer. nat.) genehmigte Dissertation. 1. Gutachter: Prof. Dr. G.-M. Greuel 2. Gutachter: Prof. Dr. B. Sturmfels Vollzug der Promotion: D 386

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3 Contents Introduction 1 1 Invariants of isolated hypersurface singularities Milnor fibration Cohomology bundle Gauss-Manin connection Brieskorn lattices Completion and (t,s)-module structure Lattices Saturation and resonance Basis representations Mixed Hodge structure Weight and Hodge filtrations Mixed Hodge structures Hodge numbers and spectral pairs Newton filtration V-filtration standard bases Algorithms for the Gauss-Manin connection Formal standard bases Formal differential deformations Matrix A of t Saturation of the Brieskorn lattice Monodromy Spectral pairs (t, s)-module structure Applications and Examples Quasihomogeneous singularities Example with Singular Example with 3 3 Jordan block

4 ii CONTENTS 3.4 Hertling s conjecture A Singular implementation 113 A.1 Singular library linalg.lib A.2 Singular library gaussman.lib Bibliography 134

5 Introduction Human thinking is based on abstraction, the concept of objects and equality. Equal objects with respect to a notion of equality form a class. Such a class of objects is considered an object itself. In mathematics, one can pass over from a given notion of equality to a coarser one by an equivalence relation on a class of objects. All objects equivalent to a given object are considered to be equal and form an (equivalence) class. This leads to a classification problem, that is the problem of describing all classes. One possible solution consists in associating to each class a normal form being an object in this class. The concept of invariants serves to approach classification problems. An invariant associates to each object an object of a possibly different type such that only one object is associated to all objects in a class. So it associates an object to each class. Of course, there is always the trivial invariant that associates to each object its class. But this does not help to solve a classification problem. A set of invariants forms an invariant and an invariant solves the classification problem if the associated object determines the class. This thesis is concerned with algorithms to compute certain invariants. The objects in this thesis are a special class of singularities. Singularities occur in all fields of mathematics and the various mathematical definitions reflect the intuitive idea of a non-smooth point of a geometrical object. The singularities in this thesis are a special class of complex space germs. A complex space is locally the zero set V(f) = {x C n f(x) = 0} C n of complex analytic functions f = f 1,...,f m and a hypersurface V(f) is a complex space defined by a single equation. It is equipped with a sheaf of local rings carrying the algebraic information of the defining equations. To consider complex spaces means to consider solution spaces of complex analytic equations as geometrical objects. A map of complex spaces is locally defined by complex analytic functions. The concept of germs allows one to consider complex spaces locally at a point. A germ of a complex space X

6 2 Introduction at a point x X is the equivalence class (X, x) of neighbourhoods of x and a hypersurface germ is the germ (X, x) of a hypersurface X. Any complex space germ is the germ V(f) = ( V(f), 0 ) (C n, 0) of the zero-set of power series f = f 1,..., f m C{x} defined in a neighbourhood of 0 C n and any hypersurface germ is the germ V(f) = ( V(f), 0 ) (C n+1, 0) of the zero-set of a single power series f C{x} defined in neighbourhood of 0 C n+1. A map of complex space germs is the equivalence class of restrictions of a map of complex spaces. A singularity is a germ (X, x) of a complex space X at a non-smooth point x X. It is called a hypersurface singularity if X is a hypersurface and it is called isolated if there are only smooth points close to x. The objects in this thesis are isolated hypersurface singularities. In chapter 1, we describe invariants of isolated hypersurface singularities. Let V(f) (C n+1, 0) be an isolated hypersurface singularity. It is defined by a power series f C{x} and one can consider f and its partial derivatives (f) = x0 (f),..., xn (f) as maps (C n+1, 0) f (C, 0), (C n+1, 0) (f) (C n+1, 0) of complex space germs. The fact that V(f) (C n+1, 0) is an isolated singularity is equivalent to the fact that 0 C n+1 is an isolated critical point of f, that is, V( (f)) = {0}. This implies that the Milnor number µ = dim C ( C{x}/ (f) ) is a finite number. Let X be the intersection of a closed ball in C n+1 centered at 0 C n+1 with the preimage under f of an open disk T in C centered at 0 C. For appropriately chosen X and T, the restriction X = X\f 1 (0) f T \{0} = T is a C fibre bundle, that is, locally at t T, the fibres are C -diffeomorphic to a product space of a neighbourhood of t and the smooth fibre X t = X f 1 (t),

7 Introduction 3 and the restriction δx f T to the boundary δx of X is a trivial fibre bundle, that is, δx is C - diffeomorphic to a product space of T and δx t. The fibre bundle f : X T is called a Milnor fibration. Different choices of X and T lead to diffeomorphic Milnor fibrations. The general fibre of the Milnor fibration is called the Milnor fibre. A parallel shift through the local product structures of the Milnor fibration over a counterclockwise loop around 0 T defines a diffeomorphism of the Milnor fibre which is trivial on the boundary. The relative isotopy class of this diffeomorphism is a topological invariant of the singularity and is called the geometrical monodromy. By J. Milnor [Mil68], the Milnor fibre is homotopy equivalent to a bouquet of µ n-spheres, that is, the space formed by µ spheres of dimension n glued at one point. This implies that the reduced (co)homology of the Milnor fibre is concentrated in dimension n and is a free Abelian group with µ generators, that is, H k (X t ) = δ k,n Z µ = Hk (Xt ) where δ is the Kronecker symbol. The ((co)homological) monodromy M is the automorphism defined by the geometrical monodromy on the (co)homology of the Milnor fibre. By the monodromy theorem, the eigenvalues of M are roots on unity and the Jordan blocks have size at most (n +1) (n +1) and at most n n for eigenvalue 1. Let M = M s M u be the decomposition of M into semisimple and unipotent part and N = log M u 2πi the logarithm of the unipotent part. Then, by the monodromy theorem, N n+1 = 0 and even N n = 0 on the generalized 1-eigenspace. The (co)homology of the smooth fibres X t, t T, form the (co)homology bundle H = t T H(X t, C). It is a flat vector bundle, that is, locally at t T, the fibres form a product space of a neighbourhood of t and the complex vectorspace H(X t, C) such that on the intersection of such neighbourhoods the isomorphism of product

8 4 Introduction spaces is independent of t. A (local) section of H is a (local) holomorphic section of the canonical projection H T. By the flatness of the cohomology bundle, differentiation of coefficient functions in the local product structures defines a derivative H t H on H, that is, it fulfills the Leibniz rule t (gv) = t (g)v + g t (v) for (local) holomorphic functions g and (local) sections v of H. In terms of a local basis of H, t = 0 is a system of µ ordinary differential equations. The flat multivalued sections of H are the solutions of t = 0 and form a µ dimensional complex vectorspace. The monodromy is defined by shifting (co)homology classes along flat sections of H over a counterclockwise loop around in 0 T. Let i : T T by the canonical inclusion. Then a flat multivalued section of H in the generalized exp( 2πiα)-eigenspace H n (X t, C) exp( 2πiα) of the monodromy corresponds to a local section of H at 0 T in the generalized α-eigenspace C α = ker(t t α) n+1 (i H ) 0 of the operator t t by an isomorphism H n (X t, C) exp( 2πiα) ψ α C α (0.1) of complex vectorspaces. The monodromy corresponds to the operator t t by (t t α) ψ α = ψ α N and hence exp( 2πit t ) ψ α = ψ α M. (0.2) An elementary section is a section of H in a generalized eigenspace of t t. The (local) Gauss-Manin connection G = α Q C{t}[t 1 ]C α is the µ-dimensional C{t}[t 1 ]-subvectorspace of (i H ) 0 generated by local elementary sections at 0 T. It is a regular C{t}[ t ]-module. By the De Rham theorem, the cohomology bundle H can be described in terms of (relative) holomorphic differential forms. Following this idea, E. Brieskorn [Bri70] defined the Brieskorn lattice H = Ω n+1 X,0 /df dωn 1 X,0.

9 Introduction 5 It is the stalk of a locally free extension of the cohomology bundle H at 0 T and a C{t}-lattice in G, that is a free C{t}-submodule of G of rank µ. The Leray residue formula implies that t [df η] = [dη] (0.3) for [df η] H. E. Brieskorn [Bri70] found the first algorithm to compute compute the complex monodromy. It has been implemented in the computer algebra system Maple V by P.F.M. Nacken [Nac90] and in the computer algebra system Singular [GPS02] by the author [Sch99, Sch02b]. Brieskorn s algorithm and the algorithms in this thesis are based on (0.2) and (0.3). The Gauss-Manin connection G has a rich structure. The generalized eigenspaces C α of the operator t t define a splitting of a decreasing filtration V of G by C{t}-lattices V α = α β C{t}C β. This filtration is called the V-filtration and tv α = V α+1. For α > 1, V α is also a C{{s}}-lattice, that is a free module of rank µ over a power series ring C{{s}} with s = t 1 and sv α = V α+1. Since [ t, t] = 1, t is a differential operator t = s 2 s on the C{{s}}-module V > 1. The regularity of the Gauss-Manin connection is reflected by the existence of the saturated, that is t t -invariant, C{t}-lattices V α. By a result of B. Malgrange [Mal74], H V > 1 and, in particular, H is a C{{s}}-lattice. In the sense of D. Barlet [Bar93, Bar00], the Brieskorn lattice H is a (t, s)-module, that is a free C{{s}}- module with an operator t such that the commutator satisfies [t, s] = s 2. In contrast to Brieskorn s algorithm which is based on the C{t}-structure, the algorithms in this thesis are based on the C{{s}}-structure. The nilpotent operator N commutes with t and s and defines a t- and s-invariant weight filtration W over Q. Multiplying the Brieskorn lattice H by powers of t resp. s defines a filtration F by C{t}-lattices F k = t k H

10 6 Introduction on G resp. a filtration F by C{{s}}-lattices F k = s k H V > 1 on V > 1. These filtrations are called Hodge filtrations. The Hodge filtration F was defined by A.N. Varchenko [Var82a] and the Hodge filtration F by J. Scherk and J.H.M. Steenbrink [SS85]. The weight resp. Hodge filtrations induce filtrations on the cohomology H n (X t, Q) resp. H n (X t, C) of the Milnor fibre. By J.H.M. Steenbrink [Ste76, SS85] and A.N. Varchenko [Var82a], the weight and Hodge filtrations define a mixed Hodge structure on the cohomology of the Milnor fibre, that is, on the graded parts of the weight filtration the induced Hodge filtrations are opposite to their complex conjugate shifted by the weight. Moreover, log M u is a morphism of mixed Hodge structures of type ( 1, 1), that is a morphism of type 2 resp. 1 with respect to the weight resp. Hodge filtrations. The Hodge numbers h p,q are the dimensions of the graded parts of the induced Hodge filtrations on the graded parts of the weight filtration, that is, dim C gr p F grw p+q Hn (X t, C) = h p,q = dim C gr p e F gr W p+q Hn (X t, C) Properties of the mixed Hodge structure lead to certain symmetries of the Hodge numbers. The spectral pairs reflect the embedding of the Brieskorn lattice H in the Gauss-Manin connection G with respect to the V- and weight filtration. They consist of µ pairs (α, l) Q Z with multiplicity dim C gr α V grw l (H /th ) = dim C gr α V grw l (H /sh ). By the isomorphisms (0.1), they correspond to the Hodge numbers inheriting their symmetries and, by (0.2), they determine the Jordan data of the complex monodromy. The first components of the spectral pairs are the spectral numbers and form the (singularity) spectrum. They consist of µ rational numbers α Q with multiplicity dim C gr α V (H /th ) = dim C gr α V (H /sh ) and they determine the eigenvalues of the complex monodromy. By M. Saito [Sai88], for Newton non-degenerate singularities, the V-filtration coincides with the Newton filtration which is defined by the Newton polyhedron of the power series f C{x}. Based on this result, S. Endrass [End02] implemented an algorithm to compute the singularity spectrum in Singular.

11 Introduction 7 By P. Deligne [Del72], there is a simultaneous splitting of the weight and Hodge filtration of a mixed Hodge structure. In particular, N is strict with respect to the weight and Hodge filtrations. By M. Saito [Sai89], this implies the existence of a C{{s}}-basis of the Brieskorn lattice H such that the basis representation of t is A 0 + sa 1 + s 2 s. The matrix A = A 0 + sa 1 of t determines the (t, s)-module structure of the Brieskorn lattice and, in particular, the spectral pairs and the complex monodromy. In chapter 2, we describe algorithms to compute, for arbitrary isolated hypersurface singularities, the complex monodromy, the spectral pairs, and the (t, s)-module structure of the Brieskorn lattice in form of M. Saito s matrices A 0 and A 1. These algorithms are based on the following results and ideas. In section 1.5, we show that the x -adic and s -adic topologies on the Brieskorn lattice coincide and that there is a C[s]-isomorphism Ĥ = C[s, x]/ (f) s C[s, x] (0.4) where Ĥ is the completion of H with respect to this topology. In section 1.10, we introduce standard bases with respect to splittings of refinements of the V-filtration. We show that the spectral pairs are the orders of a standard basis of the Brieskorn lattice with respect to a splitting of the weight refined V-filtration. We show that M. Saito s basis is a reduced standard basis with respect to a Hodge splitting of the V-filtration. In section 2.1, we develop a theory of standard bases over formal power series rings based on the idea of monomial orderings [Buc65, Buc85, GP96]. We describe a normal form and standard basis algorithm based on Buchberger s algorithm [GP96] that converges with respect to the adic topology of the power series ring. In section 2.2, we specialize this normal form algorithm to a normal form algorithm for the Brieskorn lattice using (0.4). In section 2.3, we show that it computes the matrix A = j 0 sj A j of operator t with respect to a C{{s}}- basis of the Brieskorn lattice such that A + s 2 s is the basis representation of the operator t. Here, we use the finite determinacy theorem to assume that f is a polynomial. This assumption can be replaced by appropriate degree bounds. In the following sections 2.4, 2.5, 2.6, and 2.7, we describe a sequence of C{{s}}[s 1 ]-basis transformations and

12 8 Introduction show that one compute the matrix A of t and a polynomial basis representation H of H with respect to the transformed bases. In section 2.4, the transformed basis is a C{{s}}-basis of a saturated C{{s}}-lattice and one can compute the eigenvalues of monodromy from A 1 by (0.2). Here, we use the regularity of the Gauss-Manin connection by computing the saturation of a lattice and the monodromy theorem by computing the eigenvalues of A 1 on the saturation. In section 2.4, the transformed basis is a C{{s}}-basis of a direct sum of C{{s}}C α and one can compute the complex monodromy from A 1 by (0.2). In section 2.5, the transformed basis is a C{{s}}-basis of a V α and one can compute the spectral pairs from a standard basis of H by section In section 2.5, the transformed basis is a C-basis of a direct sum α β<α Cα and one can compute M. Saito s A 0 and A 1 from a finite jet of A and a finite jet of a reduced standard basis of H by section The algorithms in this thesis compute the complex monodromy, the singularity spectrum, the spectral pairs and M. Saito s matrices A 0 and A 1. Moreover, they compute a complex basis representation of the weight and Hodge filtrations on the cohomology of the Milnor fibre. But an algorithm to compute the underlying real structure and hence the mixed Hodge structure on the cohomology of the Milnor fibre is not yet known. Appendix A contains the documentation of a Singular implementation by the author [Sch02a] of the algorithms in chapter 2. In chapter 3, we give examples and applications. We demonstrate the Singular implementation for a singularity with a 2 2 resp. 3 3 Jordan block of the monodromy. As an application, we verify Hertling s conjecture about the variance of the spectral numbers for singularities with Milnor number at most 16 using Arnold s classification [AGZV85].

13 Acknowledgments 9 Acknowledgments I wish to express my gratitude to the following people. My thesis advisors G.- M. Greuel and J.H.M. Steenbrink guided and supported me. G.-M. Greuel called my attention to Brieskorn s algorithm and J.H.M. Steenbrink to the problem of computing the spectral pairs and M. Saito s matrices A 0 and A 1. Exchange of ideas with A. Frühbis-Krüger helped me to develop the normal form algorithm for the Brieskorn lattice. H. Schönemann s advice was essential for the implementation of the algorithms in Singular. The Singular implementation was tested using examples by C. Hertling for his conjecture and by N. A Campo for the monodromy. The members of the Algebraic Geometry group of the University of Kaiserslautern supported me in many respects. I wish to thank my parents Hannelore and Volkmar Schulze for their love and care.

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15 Chapter 1 Invariants of isolated hypersurface singularities In this chapter, we describe the invariants of isolated hypersurface singularities which can be computed by the algorithms in chapter 2. The local elementary sections of the cohomology bundle at the critical value form a regular C{t}[ t ]-module, the Gauss-Manin connection. The complex monodromy can be identified with the operator t t. The description of complex cohomology in terms of differential forms by the De Rham isomorphism leads to an embedding of the Brieskorn lattice in the Gauss- Manin connection. The formal Brieskorn lattice is the completion of the Brieskorn lattice. We give an explicit description of the formal Brieskorn lattice leading to a normal form algorithm. The Gauss-Manin connection with the embedded Brieskorn lattice has a rich structure in form of the V-, weight, and Hodge filtration and a C{t}- and C{{s}}-module structure. The weight and Hodge filtration define a mixed Hodge structure on the cohomology of the Milnor fibre. This leads to symmetries of the filtrations and a close relation of the module structures in form of a certain simultaneous splitting of the V- and Hodge filtration. The relation of the filtrations is reflected by the spectral pairs corresponding to the Hodge numbers and the relation of the two module structures by M. Saito s matrices A 0 and A 1. These invariants can be computed by the algorithms in chapter 2. Standard bases are well known in computer algebra and occur naturally in the context of filtrations and splittings. We introduce standard bases with respect to the V-filtration and describe the invariants in terms of standard bases.

16 12 1. Invariants of isolated hypersurface singularities 1.1 Milnor fibration In this section, we introduce the Milnor fibration associated to an isolated hypersurface singularity. It is a fibre bundle formed by the smooth fibres near the singular fibre. Its general fibre is homotopy equivalent to a bouquet of spheres of dimension equal to the dimension of the singularity. The number of spheres is the Milnor number of the singularity. The construction and results in this section can be found in [Mil68] and [Loo84]. We denote row vectors by a lower bar and column vectors by an upper bar. Lower indices are row indices and upper indices are column indices. Let V(f) (C n+1, 0) be an isolated hypersurface singularity. Then (C n+1, 0) f (C, 0) is the germ of a holomorphic function with isolated critical point 0 C n+1. Let x = (x 0,...,x n ) be a local coordinate system at 0 C n+1 and := ( 0,..., n ) where j := xj. Let t be a local coordinate at 0 C. Then the fact that f has an isolated critical point is equivalent to V( (f)) = {0} = V(x) and hence to (f) = x by the analytic Nullstellensatz [djp00, Thm ]. This implies that there is a k 1 such that x k (f). Definition The dimension 1 µ := dim C ( C{x}/ (f) ) < of the Jacobian algebra C{x}/ (f) is called the Milnor number. An isolated hypersurface singularity is determined by a finite jet of the defining power series. Theorem (Finite determinacy theorem). Let f x C{x} with x k+1 x 2 (f). Then f is k-determined, that is, for any g C{x} with f g x k+1, there is an automorphism φ Aut C{x} such that φ(f) = g. In particular, if k µ + 1 then f is k-determined. Proof: [djp00, Thm ] Let D 2n+2 be the open unit ball and S 2n+1 the unit sphere in C n+1. By the curve selection lemma [Mil68, Lem. 3.1], we can choose δ > 0 such that f is defined on δd 2n+2 with only critical point 0 and the singular fibre f 1 (0) intersects δ S 2n+1, δ < δ, transversely. Since S 2n+1 is compact, we

17 1.1. Milnor fibration 13 can choose ǫ > 0 such that the fibre f 1 (t), t ǫd 2, intersects δs 2n+1 transversely. Finally, we set T := ǫd 2, X := δd 2n+2 f 1 (T), T := T \{0}, X := X\f 1 (0), and denote by i : T T the inclusion. Then X f T is a good representative of f in the sense of [Loo84, 2.7]. We denote by the fibre of f over t T and by X t := X f 1 (t) X U := X f 1 (U) the preimage of an open subset U T under f. By the Ehresmann fibration theorem, X f T is a C fibre bundle and, by [Loo84, Prop. 2.9], it is independent of the choices of ǫ and δ. Definition The C fibre bundle f : X T is called the Milnor fibration and the general fibre X t, t T, is called the Milnor fibre. Figure 1.1 shows the Milnor fibration. J. Milnor studied the Milnor fibration and proved the following theorem. Theorem The Milnor fibre is homotopy equivalent to the bouquet of µ n-spheres. Proof: [Mil68, Thm. 7.2] Recall that the bouquet of a set of pointed topological spaces is the topological space obtained by gluing these spaces at their base points. Corollary The reduced (co)homology of the Milnor fibre is concentrated in the dimension of the singularity, that is, where δ is the Kronecker symbol. H k (X t ) = δ k,n Z µ = Hk (Xt )

18 14 1. Invariants of isolated hypersurface singularities Figure 1.1: The Milnor fibration 0 X X 0 X t f 0 t T 1.2 Cohomology bundle In this section, we introduce the (co)homology bundle and the (co)homological monodromy. The (co)homology of the fibres of the Milnor fibration form a local system with a natural flat connection on the sheaf of holomorphic sections. Shifting (co)homology classes along flat sections around the singular fibre defines an automorphism of the (co)homology of the Milnor fibre, the (co)homological monodromy. By the monodromy theorem, its eigenvalues are roots of unity and the size of its Jordan blocks is bounded by the dimension of the singularity added by one. An introduction to local systems, (flat) connections, and monodromy can be found in [Del70]. In the following, we denote the n-th homology resp. cohomology by H n

19 1.2. Cohomology bundle 15 resp. H n and omit the index n if the statement applies to homology and cohomology. Since f : X T is a C fibre bundle, H(U) := H(X U ) defines a Z-sheaf H. Note that H n Z C = R n f C X. The O T -sheaf H := O T ZT H is the sheaf of holomorphic sections of H. By corollary 1.1.5, H is a locally free Z-sheaf of rank µ and hence H is a locally free O T -sheaf of rank µ. Note that H Z C is a local system in the sense of [Del70]. Definition The Z-sheaf H is called the (co)homology sheaf and the O T -sheaf H is called the (co)homology bundle. We denote by (Ω, d) the complex of sheaves of holomorphic differential forms and by T the sheaf of holomorphic vector fields. A connection on H is a map which fulfills the Leibniz rule H Ω 1 T O T H (gv) = d(g) v + g (v) for (local) sections g Γ(U, O T ) and v Γ(U, H ). A (local) section v Γ(U, H ) is called a flat section if (v) = 0 and the sheaf ker( ) of flat sections is a locally constant C-sheaf. The covariant derivative H X H of along a (local) vector field X Γ(U, T T ) is defined by It fulfills the Leibniz rule X (v) = (v), X. X (g v) = X(g) v + g X (v) for (local) sections g Γ(U, O T ) and v Γ(U, H ). A connection on H extends to a map of complexes Ω T O T H Ω T [1] O T H

20 16 1. Invariants of isolated hypersurface singularities by (ω v) = dω v + ( 1) k (v) for (local) sections ω Γ(U, Ω k T ) and v Γ(U, H ). If 2 = 0 or equivalently then is called a flat connection. [X,Y ] = [ X, Y ] Definition The flat connection on H with ker = H is called the Gauss-Manin connection. Note that the Gauss-Manin connection H Ω 1 T O T H is defined by (gv) := dg v for (local) sections g Γ(U, O T ) and v Γ(U, H). A connection on H defines a dual connection on H by X (φ)(v) = X(φ(v)) φ( X(v)) for (local) sections φ Γ(U, H ) and v Γ(U, H ) and (local) vector fields X Γ(U, T T ). Note that the homological and cohomological Gauss-Manin connections are dual connections. We denote the covariant derivative of along t on H and i H by t := t. Let T u T τ exp(2πiτ) be the universal covering of T where τ is a coordinate on T C. Definition The pullback X := X T T is called the canonical Milnor fibre.

21 1.2. Cohomology bundle 17 Then is a C fibre bundle with X τ X T = X u(τ). Since T is contractible, H(U) := H(X U ) defines a free Z-sheaf H of rank µ on T and u H is the sheaf of flat multivalued sections of H. For a field extension Q K, we denote by H K := H Z K the corresponding extension of scalars. We consider A H as a global flat multivalued section A(t) of H. Note that t A(t) = 0 for A H. Lifting closed paths in T along section in H defines the monodromy representation π 1 (T, t) Aut(H t ) M t on H t where is the counterclockwise generator. The monodromy representations on the H t induce the monodromy representation π 1 (T ) Aut(H) M on H where is the counterclockwise generator such that (Ms)(τ) = s(τ + 1) for s H. The sheaf H is determined by the monodromy representation up to isomorphism. Definition The automorphism M Aut(H) is called the ((co)homological) monodromy. Note that the homological monodromy is dual to the inverse of the cohomological monodromy. Figure 1.2 shows the monodromy. The most important result on the monodromy is the following theorem.

22 18 1. Invariants of isolated hypersurface singularities Figure 1.2: The (geometrical) monodromy 0 M X X 0 X t f 0 t T Theorem (Monodromy theorem). The eigenvalues of M are roots of unity, the Jordan blocks have size at most (n + 1) (n + 1), and the Jordan blocks with eigenvalue 1 have size at most n n. Proof: [Bri70] and others. From now on, we denote the homology resp. cohomology sheaf by H n resp. H n and the homology resp. cohomology bundle by H n resp. H n. 1.3 Gauss-Manin connection In this section, we introduce the (local) Gauss-Manin connection. The local elementary sections of the cohomology bundle at the critical value generate a

23 1.3. Gauss-Manin connection 19 regular C{t}[ t ]-module, the (local) Gauss-Manin connection. It has a C{t}- and C{{s}}-module structure and the operator t t corresponds to the complex monodromy. The construction in this section can be found in [AGZV88] and [Her93, Her00]. Let M = M s M u = M u M s be the decomposition of M into semisimple part M s and unipotent part M u and N := log M u End C (H C ). 2πi By theorem 1.2.5, the eigenvalues of M s are roots of unity. Note that 2πiN End Q (H Q ) is defined over Q. Let H C = λ H λ C be the decomposition of H C into generalized eigenspaces H λ C := ker(m s λ) of M and M λ := M H λ C. Let H 1 C := λ 1 Hλ C that the decomposition H C = HC 1 H 1 C By theorem 1.2.5, For α Q, we denote H Q = H 1 Q H 1 Q. N n+1 = 0, ( ) n N H 1 = 0. C λ α := exp( 2πiα). and H 1 Q := H 1 C H Q. Note is defined over Q, that is, Lemma For A H λα C, the multivalued section s α(a) of H n defined by s α (A)(t) := t α exp(n log t)a(t) is single-valued. Proof: Since M(s α (A))(t) = t α exp(2πiα) exp(n log t) exp(2πin)ma(t) = t α λ 1 α exp(n log t)m 1 u M s M u A(t) = t α exp(n log t)a(t) = s α (A)(t)

24 20 1. Invariants of isolated hypersurface singularities Figure 1.3: An elementary section s α (A) 0 0 H t A H t t 0 t the section s α (A) is M-invariant and hence single-valued. Note that the twist t α exp(n log t) is inverse to the action of the monodromy on H λα C. Definition A section s α (A) is called an elementary section. Figure 1.3 shows an elementary section. The flat multivalued section s α (A) defined by A H λα C is twisted by the factor tα exp(n log t) depending on t such that it glues over 1 to a global single-valued section. Let D be the sheaf of holomorphic differential operators. Note that D T,0 = C{t}[ t ] with [ t, t] = 1. The elementary sections generate a D T -module G := i s α ( H λ α C ) α Q OT i H n. ( Note that O T s α H λ α ) C and hence G is independent of the coordinate t. Definition We call the D T,0 -module G := G 0 the (local) Gauss-Manin connection. Figure 1.4 shows the (local) Gauss-Manin connection. The eigenvalue level of the operator t t is on the vertical axis Q. We denote the finite dimensional C-vectorspaces C α by horizontal line segments and consider their lengths as the corresponding dimensions. We denote operators by arrows and submodules by dashed lines.

25 1.3. Gauss-Manin connection 21 Figure 1.4: The (local) Gauss-Manin connection 6 α α α α α α α α 2 C α+7 C α+6 C α+5 C α+4 C α+3 C α+2 C α+1 C α t t t t t t t t t t t t t t N N N N N N N N G G Since t α exp(n log t) is invertible, is an inclusion with image H λα ψ α C G A (i s α (A)) 0 C α := im ψ α. Note that, for an N-invariant subspace F C α, C{t}F is independent of the coordinate t. Lemma t ψ α = ψ α+1 2. t ψ α = ψ α 1 (α + N) 3. (t t α) ψ α = ψ α N

26 22 1. Invariants of isolated hypersurface singularities 4. exp( 2πit t ) ψ α = ψ α M λα Proof: Equality 1 and 2 follow from the definition of ψ α. Combining equalities 1 and 2 gives equality 3. Applying the exponential to equality 3 gives equality 4. Note that, by lemma , C α = ker(t t α) n+1 is the generalized α-eigenspace of t t on (i H ) 0. From now on, we identify the operator N on H λα C with the operator N : C α ψα H λα C N H λα C ψ α C α on C α such that t t α = N. Corollary The operator is bijective and [t, N] = 0. C α t C α+1 2. The operator C α t is bijective for α 0 and [ t, N] = 0. C α+1 3. C α = im ψ α = ker(t t α) n+1 4. G is a free O T [t 1 ]-module of rank µ. Proof: This follows from corollary and lemma By lemma , the operators t and t are related by t = t 1 (α + N). This relation generalizes to a relation of their powers. Lemma As C-homomorphisms C α+k C α, k t = t k k (α + j + N). j=1

27 1.3. Gauss-Manin connection 23 Proof: By lemma and corollary , t k = = k t 1 t t j=1 k t 1 (α + j + N) j=1 k = t k (α + j + N). j=1 Figure 1.5: The V-filtration on the Gauss-Manin connection α α α 1 t t s s V α+2 V α+1 V α 2 Definition The V-filtration V = (V α ) α Q on G is the decreasing

28 24 1. Invariants of isolated hypersurface singularities filtration by C{t}-modules V α := α β V >α := α<β C{t}C β = C{t}C β = α β<α+1 α<β α+1 C{t}C β, C{t}C β. Figure 1.5 shows the V-filtration on the Gauss-Manin connection. Note that the V α and V >α are independent of the coordinate t. The generalized eigenspaces C α define a canonical splitting C α = V α /V >α of the V-filtration and one can consider C α as a subspace and a subquotient of G. For a filtration F on G, we denote by FC α the induced filtration on C α considered as a subquotient of G. Proposition The V α and V >α are free C{t}-modules of rank µ. Proof: This follows from corollary and By corollary , the inverse of the operator t is defined on V > 1. It extends to a module structure over a power series ring. Definition The ring of microdifferential operators with constant coefficients is defined by { C{{s}} := a k s k C[s] k=0 k=0 Note that C{{s}} is a discrete valuation ring. a } k k! tk C{t}. Lemma For α > 1 or α / Z, C{t}t α is a free C{{s}}-module of rank 1 with s := t 1 := t dt. 0 Proof: Since a k t k t α = k=0 k=0 j=1 k (α + j)a k t k t α, the claim is equivalent to the fact that k=0 a kt k C{t} if and only if k=0 Q k j=1 (α+j) k! a k t k C{t}. Proposition For α > 1 or α / Z, C{t}C α is a free C{{s}}-module of rank dim C C α with s := t 1. In particular, for α > 1 resp. α 1, V α resp. V >α is a free C{{s}}-module of rank µ.

29 1.3. Gauss-Manin connection 25 Proof: Since N is nilpotent on C α, dim C N(C α ) < dim C C α. By induction on dim C C α and by corollary and , C{t}N(C α ) = N(C{t}C α ) is a free C{{s}}-module of rank dim C C α. By lemma and , (C{t}t α ) dim C(C α /N(C α )) =C{{s}} C{t}(C α /N(C α )) =C{{s}} (C{t}C α )/(C{t}N(C α )) is a free C{{s}}-module of rank dim C C α dim C N(C α ) and hence C{t}C α is a free C{{s}}-module of rank dim C C α. Then the claim follows from corollary Definition We denote s := 1 t Definition We call the maximal C{{s}}-module G := C{t}C 0 C{t}[t 1 ]C α 1<α<0 in G the reduced (local) Gauss-Manin connection. The µ-dimensional C{{s}}[s 1 ]-vector space G C{{s}} C{{s}}[s 1 ] is called the Gauss-Manin system [Pha79, SS85]. Note that, for α > 1, V α = V α G. The following lemma, summarizes the basic properties of the V-filtration. Lemma The generalized eigenspaces C α define a canonical splitting of the V-filtration. 2. The operators C α = V α /V >α are bijective. V α and V >α t V α+1, t V >α+1

30 26 1. Invariants of isolated hypersurface singularities 3. The operator V α resp. V >α is bijective for α > 0 resp. α 0. t t V α 1 V >α 1 4. For α > 1 resp. α 1, the operator is bijective. V α resp. V >α s V α+1 s V >α+1 Proof: This follows from corollary and The operator t is a differential operator with respect to the C{{s}}-structure. Definition We denote Note that t t = s s. Lemma s := 2 t t [ s, s] = 1 In particular, t = s 2 s is a differential operator with respect to the C{{s}}- structure. Proof: Since [ t, t] = t t t t = 1, [t, s] = ts st = t 1 t 1 t t = t 1 ( t t t t ) t 1 = t 2 = s 2. and hence [ t 2 t, s] = 1. The C{t}[ t ]-module structure of the Gauss-Manin connection can be described in terms of the Jordan data of the monodromy. Proposition Let n λ,j, j = 1,..., m λ, be the Jordan blocks sizes of M λ and 1 α λ < 0 with λ = exp(2πiα λ ). Then there is a C{t}[ t ]- isomorphism G m λ = C{t}[ t] C{t}[ t ]/C{t}[ t ](t t α λ ) n λ,j. λ j=1

31 1.4. Brieskorn lattices 27 Proof: Let be a decomposition of HC λ dim C H λ,j such that HC λ m λ = j=1 H λ,j C C and C α,j := ψ α ( H λ,j C into Jordan blocks Hλ,j C of M λ of size n λ,j = V 1 = λ m λ j=1 ) C{t}C α λ,j. Let A λ,j H λ,j C be an N-cyclic vector. By corollary , t : C α C α 1 is bijective for α 0 and hence as a C{t}[ t ]-module. G = C{t} [ t 1] V 1 m λ = λ j=1 n λ,j 1 k=0 m λ n λ,j 1 C{t} [ t 1] s αλ (N k A λ,j ) = C{t}[ t ](t t α λ ) k s αλ (A λ,j ) λ j=1 k=0 m λ = C{t}[ t ]/C{t}[ t ](t t α λ ) n λ,j λ j=1 1.4 Brieskorn lattices In this section, we introduce the Brieskorn lattices. By the De Rham isomorphism, the cohomology bundle can be described in terms of (relative) holomorphic differential forms. This defines locally free extensions of the cohomology bundle, the Brieskorn lattices. The (local) Brieskorn lattices are embedded in the (local) Gauss-Manin connection as C{t}- and C{{s}}-lattices. The C{t}[ t ]-module structure of the (local) Gauss-Manin connection can be expressed in terms of holomorphic differential forms on the (local) Brieskorn lattices. The results in this section can be found in [Bri70, Seb70, Mal74]. We denote by (Ω, d) the complex of sheaves of holomorphic differential forms. Since the Milnor fibre X t, t T, is a Stein complex manifold, the De Rham homomorphism H n DR (Ω X t ) ρ t H n (X t, C) = Hom C (H n (X t, C), C)

32 28 1. Invariants of isolated hypersurface singularities defined by is an isomorphism. We denote by ρ t ( [ω] ) (δ) = (Ω X/T, d) = (Ω X/df Ω 1 X, d) the complex of sheaves of relative holomorphic differential forms with respect to f : X T. The isomorphisms ρ t glue to a natural isomorphism δ ω H n (f Ω X /T ) ρ H n. E. Brieskorn [Bri70] defined the following extensions of H n. Definition The O T -modules H := H n (f Ω X/T ) = Hn (f Ω X /df f Ω X ) H := f Ω n X/T /d(f Ω n 1 X/T ) = f Ω n X /(d(f Ω n 1 X ) + df f Ω n 1 X ) H := f Ω n+1 X /df d(f Ω n 1 X ) are called the Brieskorn lattices. We call their stalks H := H 0, H := H 0, H := H 0 at 0 T the (local) Brieskorn lattices. We refer to H resp. H as the (local) Brieskorn lattice. The following O T -module occurs naturally in the context of Brieskorn lattices. Definition Ω := f Ω n+1 X /df f Ω n X. The operators d and df define the following exact sequences. Lemma (Poincaré lemma). 0 C O X d Ω 1 X d d Ω n+1 X 0 is an exact sequence of C-vectorspaces.

33 1.4. Brieskorn lattices 29 Lemma (De Rham lemma). 0 O X df Ω 1 X df df Ω n+1 X Ω 0 is an exact sequence of O X -modules. Theorem H, H, and H are locally free O T -modules of rank µ and H = H n (Ω X/T,0 ) = Hn (Ω X,0 /df Ω X,0 ), H = Ω n X/T,0 /dωn 1 X/T,0 = Ωn X,0 /(dωn 1 X,0 + df Ωn 1 X,0 ), H = Ω n X,0/df dω n 1 X,0 are their stalks at The identity and df induce O T -inclusions H H df H (1.4.1) restricting to O T -isomorphisms ρ H n H T Proof: 1. [Bri70] and [Seb70, Cor.1] 2. This follows from lemma H T df H T (1.4.2) By theorem , there is an embedding of H, H, and H in i H n compatible with the inclusions (1.4.1). We consider H, H, and H as O T -submodules of i H n. Proposition The identity induces an O T -isomorphism H /H Ω. 2. The differential d induces an O T -isomorphism H /H d Ω.

34 30 1. Invariants of isolated hypersurface singularities Proof: 1. This follows from the definition. 2. This follows from lemma By the isomorphisms (1.4.2), (local) sections η Γ ( ) U, f Ω n X and ω Γ ( ) ( ) U, f Ω n+1 X define (local) sections s [η] Γ(U, H n ) and s ( [ω] ) Γ(U, H n ) such that s ( [η] ) ( (t) = ρ t [η Xt ] ), s ( [ω] ) ([ ω (t) = ρ t ]). df Xt Definition For (local) sections η Γ ( U, f Ω n X ) and ω Γ ( U, f Ω n+1 X ), the (local) sections s ( [η] ) and s ( [ω] ) are called geometrical sections. The action of the derivative t on the Brieskorn lattices can be expressed in terms of holomorphic differential forms and the operators d and df. Proposition H [η] t H ] [ dη df [df η] t H [dη] Proof: We follow the proof by E. Brieskorn [Bri70, Satz 1]. Let η Γ ( ) U, f Ω n+1 X a (local) section and δ Γ(U, Hn ) a flat (local) section. Let H n (X t, C) H n+1 (X\X t, C) be the Leray coboundary. By shrinking U, we may assume that there is a (δ) H n+1 (X\U, C) inducing (δ(t)) for all t U. Then the Leray residue

35 1.4. Brieskorn lattices 31 formula implies that ( t s ( [η] )) (δ) = t ( s ( [η] ) (δ) ) s ( [η] )( t δ ) = t = 1 2πi t = 1 2πi = 1 2πi = 1 2πi = 1 2πi = δ (δ) (δ) (δ) (δ) df η f t df η t f t df η (f t) 2 ( dη f t d η ) f t (δ) dη f t dη ([ dη df = s df ]) (δ) and hence t [η] = [ dη df ] and t [df η] = [dη]. The bijectivity follows from lemma By proposition 1.4.8, the derivative t has a pole on each of the Brieskorn lattices. The pole orders are the same and at most equal to the dimension of the singularity added by one. Corollary The minimal κ with f κ (f) equals the minimal κ with t κ t H H, resp. t κ t H H, resp. t κ t H H. δ η 2. Proof: 1 κ n + 1 and κ = 1 if and only if the singularity V(f) is quasihomogeneous. 1. Since 0 C n+1 is an isolated critical point of f, f x and there is a k 0 such that x k (f). Hence, there is a minimal 1 κ < with f κ (f).

36 32 1. Invariants of isolated hypersurface singularities We denote dx = dx 0 dx n and, for 0 j n, dx b j = dx 0 dx j 1 dx j+1 dx n. Since f k Ω n+1 X,0 = fk C{t}dx = f k dx, n n df Ω n X,0 = df C{t}dx b j = C{t} j (f)dx = (f) dx, j=0 j=0 κ is minimal with f κ dx df Ω n X,0. By proposition 1.4.8, κ is minimal with t κ H t 1 H or equivalently t κ t H H. Then the claim follows from proposition By J. Briançon and H. Skoda [BS74], κ n+1 and this bound is strict. By K. Saito [Sai71], f (f) if and only if the singularity V(f) is quasihomogeneous. The geometrical sections have moderate growth with respect to the flat multivalued sections and the geometrical sections in H tend to 0 as t T tends to 0 T. Theorem H G 2. H V > 1. Proof: 1. [Bri70, Satz 2] 2. [Mal74, lem. 4.5] P. Pham [Pha77, Pha79] proved the following result in the context of the Gauss-Manin system. Proposition H, H, and H are free C{{s}}-modules of rank µ.

37 1.5. Completion and (t,s)-module structure H s H s H [dη] [df η] [ dη df ] [η] Proof: By theorem and proposition 1.4.8, sh = 1 t H = H H and hence C[s]H H. By theorem , H V > 1 and, by proposition , V > 1 is a free C{{s}}-module. Since H and V > 1 are C{t}- lattices, there is a k 0 such that s k H s k V > 1 = V >k 1 = t k V > 1 H and, by proposition , V >k 1 is a C{{s}}-module. Hence, H is a free C{{s}}-module. By proposition , H /sh = H /H = Ω0 and dim C Ω 0 = µ. By Nakayama s lemma, this implies that H is a free C{{s}}-module of rank µ. Then the claim follows from proposition From now on, we abbreviate Ω := Ω 0, Ω := Ω X, Completion and (t,s)-module structure In this section, we consider the completions of the Brieskorn lattices. By E. Brieskorn [Bri70], the x -adic and t -adic topologies on the Brieskorn lattices coincide. We show that the x -adic and s -adic topologies coincide on the Brieskorn lattices. The formal Brieskorn lattices are the completions of the Brieskorn lattices with respect to this topology. We give an explicit description of the formal Brieskorn lattice leading to the normal form algorithm in section 2.2. Definition We call the x -adic completions Ĥ = H n ( Ω /df Ω ), Ĥ = Ω n /d Ω n 1 + df Ω n 1, Ĥ = Ω n+1 /df d Ω n 1

38 34 1. Invariants of isolated hypersurface singularities of the Brieskorn lattices the formal Brieskorn lattices. We refer to Ĥ as the formal Brieskorn lattice. The following proposition is essential for Brieskorn s algorithm to compute the complex monodromy. Theorem The t -adic and x -adic topology on H, H, resp. H coincide. In particular, the t -adic completion of H, H, resp. H is naturally isomorphic to Ĥ, Ĥ, resp. Ĥ. Proof: [Bri70, Prop. 3.3] Corollary Ĥ, Ĥ, and Ĥ are free C[t]-modules of rank µ. Proof: Since completion is faithfully flat, this follows from proposition and theorem Proposition The s -adic and x -adic topology on H, H, resp. H coincide. In particular, the s -adic completion of H, H, resp. H is naturally isomorphic to Ĥ, Ĥ, resp. Ĥ. Proof: Let [ g i (f) j (f)dx ] ( (f) 2k dx + df dω n 1)/ df dω n 1 H. Then, by proposition , [ g i (f) j (f)dx ] = [ ( 1) i df ( g j (f)dx b i)] = s [ ( 1) i d ( g j (f)dx b i = s [ ( i g j (f) ) dx ] )] = s [( i (g) j (f) + g i j (f) ) dx ] s (( (f) 2(k 1) dx + df dω n 1)/ df dω n 1) and hence by induction ( (f) 2k dx + df dω n 1)/ df dω n 1 s k H. Since 0 is an isolated critical point of f, there is an m 1 such that x (f) x m and hence ( x 2km dx + df dω n 1)/ df dω n 1 ( (f) 2k dx + df dω n 1)/ df dω n 1.

39 1.5. Completion and (t,s)-module structure 35 This implies that ( x 2km dx + df dω n 1)/ df dω n 1 s k H ( x k dx + df dω n 1)/ df dω n 1. Hence, s -adic and x -adic topology on H coincide. Then the claim follows from proposition Corollary Ĥ, Ĥ, and Ĥ are free C[s]-modules of rank µ. 2. Ĥ s Ĥ s Ĥ [dη] [df η] [ dη df ] [η] Proof: Since completion is faithfully flat, this follows from proposition and The following description of the formal Brieskorn lattice leads to the normal form algorithm in section 2.2. Proposition There is a C[s]-isomorphism Proof: Since there is a natural map Ĥ =C[[s]] C[s, x]/ (f) s C[s, x]. df d Ω n 1 = (df sd)d Ω n 1 (df sd) Ω n [s] Ĥ ι Ωn+1 [s]/(df sd) Ω n [s]. By corollary , ι is a C[s]-homomorphism. Let ω = k=0 ω ks k Ω n [s] with (df sd)ω Ω n+1. Then df ω k+1 = dω k and hence, by proposition , s[dω k+1 ] = [df ω k+1 ] = [dω k ] Ĥ for all k 0. Then [dω 0 ] k 0 sk Ĥ = {0} and hence, by definition of H, dω 0 df d Ω n 1 = d(df Ω n 1 ).

40 36 1. Invariants of isolated hypersurface singularities By lemma 1.4.3, ω 0 d Ω n 1 + df Ω n 1 and hence This implies that (df sd)ω = df ω 0 df d Ω n 1. (df sd) Ω n [s] Ω n+1 = df d Ω n 1 and hence ι is injective. By lemma 1.4.3, d Ω n = Ω n+1 and hence, by corollary , ι is surjective. Note that ι is inverse to the canonical C[s]-projection Ω n+1 [s]/(df sd) Ω n [s] π Ωn+1 /df d Ω n 1. Since Ω n [s] is a free C[s, x]-module of rank 1 with generator dx, there is a C[s, x]-isomorphism C[s, x] For η = n j=0 ( 1)j g j dx b j Ω n [s], dx Ωn+1 [s]. (df sd)η = n ( j (f)g j s j (g j ))dx = ( (f) s )gdx j=0 and hence dx induces a C[s]-isomorphism C[s, x]/ (f) s C[s, x] dx Ωn+1 [s]/(df sd) Ω n [s]. D. Barlet [Bar93, Bar00] considered the following algebraic structure. Definition A (t,s)-module is a free C[s]-module of finite rank endowed with a C-endomorphism t fulfilling [t, s] = s 2. Corollary Ĥ, Ĥ, and Ĥ are (t,s)-modules. Proof: This follows from lemma and corollary Lattices In this section, we consider C{t}- and C{{s}}-lattices in the Gauss-Manin connection. A C{t}-lattice in the Gauss-Manin connection is a free C{t}- submodule of rank µ. We call a free C{{s}}-submodule of rank µ a C{{s}}- lattice. The V-filtration consists of C{t}- and C{{s}}-lattices and the Brieskorn lattices are C{t}- and C{{s}}-lattices.

41 1.6. Lattices 37 Definition We call a free C{t}- resp. C{{s}}-submodule of G resp. G of rank µ a C{t}- resp. C{{s}}-lattice. We call a C{t}- resp. C{{s}}-lattice a lattice. From now on, we denote by V the V-filtration on G resp. G. Remark By proposition and , the V α are C{t}- and C{{s}}-lattices. 2. By proposition and , H, H, and H are C{t}- and C{{s}}-lattices. Lemma Let L be a C{t}- resp. C{{s}}-lattice. Then there are α, β Q such that V α L V β. Proof: Let L and L be two C{t}- resp. C{{s}}-lattices. Since C{t} resp. C{{s}} is a discrete valuation ring, there is a k 0 such that t k L L resp. s k L L. By lemma and proposition and , this implies the claim. Lemma A C{t}-lattice L is a C{{s}}-lattice if and only if sl L. 2. A C{{s}}-lattice L is a C{t}-lattice if and only if tl L. Proof: This follows from lemma as in the proof of proposition Saturation and resonance In this subsection, we consider saturated and non-resonant lattices. The V-filtration consists of saturated non-resonant lattices. By the regularity of the Gauss-Manin connection, the saturation of a lattice by the operator t t = s s 1 is a lattice. The complex monodromy on the cohomology of the Milnor fibre corresponds to this operator on L/tL = L/sL for a saturated non-resonant lattice L. Definition Let L be a C{t}- resp. C{{s}}-lattice. 1. If t t L = (s s 1)L L then L is called saturated. 2. If L is saturated then the endomorphism res L End C (L/tL) resp. res L End C (L/sL) induced by t t = s s 1 is called the residue of L.

42 38 1. Invariants of isolated hypersurface singularities 3. If res L has non-zero integer differences of eigenvalues then L is called resonant. Note that the residue is independent of the coordinate t. Remark By lemma and , the V α are saturated lattices. 2. By corollary , H, H, and H are saturated if and only if the singularity V(f) is quasihomogeneous. Lemma Let L be a saturated lattice. 1. If L is a C{t}- and C{{s}}-lattice then 2. If V α L V β then ( L = tl = sl. α γ<β and the L C γ are N-invariant. ) L C γ V β 3. If the eigenvalues of res L are in [α, β] resp. [α, β) then Proof: V α L V >β 1 resp. V α L V β Since L is saturated and [ t, t] = 1, s 1 tl = t tl L and hence tl sl. By Nakayama s lemma, and hence tl = sl. dim C (L/tL) = µ = dim C (L/sL) 2. Since V α L V β, L = L V α = (L α γ<β C γ ) V β. Since L is saturated, L α γ<β Cγ is t t -invariant and L C γ is the γ-eigenspace of t t. Hence, by lemma , L C γ is N-invariant.

43 1.6. Lattices Since L C γ is the γ-eigenspace of t t and res L is induced by t t, this follows from 2. Definition Let L be a lattice, L 0 := L, and L k := k (t t ) j L = L k 1 + t t L k 1 j=0 for k 1. Then L := k 0 L k is called the saturation of L. Lemma Let L be a lattice. Then L k and L are lattices. Proof: We may assume that L is a C{t}-lattice. By induction on k, we may assume that L k 1 is a C{t}-lattice. Let g C{t} and v L k 1. Then gt t v = t t gv t t (g)v t t L k 1 + L k 1 = L k. Hence, C{t}L k L k and L k is a C{t}-lattice. By lemma 1.6.3, there is an α Q such that L V α. Since V α is saturated, L k V α for all k 0. Since C{t} is Noetherian and V α a finite C{t}-module, V α is Noetherian and hence the sequence of C{t}-submodules L 0 L 1 L 2 V α is stationary. Hence, there is a k 0 such that L = L k and L is a C{t}- lattice. The following bound for the minimal k 0 with L k = L is due to R. Gérard and A.H.M. Levelt [GL73, Thm. 4.2]. We give an elementary proof. Proposition Let L be a lattice. Then L = L µ 1 Proof: We may assume that L is a C{t}-lattice. By lemma 1.6.3, there is an α Q such that L V α. Let e = (e 1,...,e µ ) be a Jordan C-basis of α β<α+1 C β = V α /tv α